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Brocard circle

In geometry, the Brocard circle (or seven-point circle) for a triangle is a circle defined from a given triangle. It passes through the circumcenter and symmedian of the triangle, and is centered at the midpoint of the line segment joining them (so that this segment is a diameter).

In terms of the side lengths $a$, $b$, and $c$ of the given triangle, and the areal coordinates$(x,y,z)$ for points inside the triangle (where the $x$-coordinate of a point is the area of the triangle made by that point with the side of length $a$, etc), the Brocard circle consists of the points satisfying the equation^{[1]}

The two Brocard points lie on this circle, as do the vertices of the Brocard triangle.^{[2]}
These five points, together with the other two points on the circle (the circumcenter and symmedian), justify the name "seven-point circle".

If the triangle is equilateral, the circumcenter and symmedian coincide and therefore the Brocard circle reduces to a single point.^{[4]}

History

The Brocard circle is named for Henri Brocard,^{[5]} who presented a paper on it to the French Association for the Advancement of Science in Algiers in 1881.^{[6]}